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Difference between revisions of "Bott periodicity theorem"

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is the space of loops on  $  X $,  
 
is the space of loops on  $  X $,  
 
and  $  \sim $
 
and  $  \sim $
is weak homotopy equivalence, in particular  $  \pi _ {i} (U) = \pi _ {i+2} (U) $
+
is [[weak homotopy equivalence]], in particular  $  \pi _ {i} (U) = \pi _ {i+2} (U) $
 
for  $  i = 0, 1 \dots $
 
for  $  i = 0, 1 \dots $
 
where  $  \pi _ {i} $
 
where  $  \pi _ {i} $
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====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> R. Bott, "The stable homotopy of the classical groups" ''Ann. of Math. (2)'' , '''70''' : 2 (1959) pp. 313–337 {{MR|0110104}} {{ZBL|0129.15601}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> J.W. Milnor, "Morse theory" , Princeton Univ. Press (1963) {{MR|0163331}} {{ZBL|0108.10401}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> M.F. Atiyah, "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017220/b01722032.png" />-theory: lectures" , Benjamin (1967) {{MR|224083}} {{ZBL|}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> D. Husemoller, "Fibre bundles" , McGraw-Hill (1966) {{MR|0229247}} {{ZBL|0144.44804}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> J.C. Moore, "On the periodicity theorem for complex vector bundles" , ''Sem. H. Cartan'' (1959–1960)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> M.F. Atiyah, R. Bott, "On the periodicity theorem for complex vector bundles" ''Acta Math.'' , '''112''' (1964) pp. 229–247 {{MR|0178470}} {{ZBL|0131.38201}} </TD></TR></table>
+
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> R. Bott, "The stable homotopy of the classical groups" ''Ann. of Math. (2)'' , '''70''' : 2 (1959) pp. 313–337 {{MR|0110104}} {{ZBL|0129.15601}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> J.W. Milnor, "Morse theory" , Princeton Univ. Press (1963) {{MR|0163331}} {{ZBL|0108.10401}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> M.F. Atiyah, "$K$-theory: lectures" , Benjamin (1967) {{MR|224083}} {{ZBL|}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> D. Husemoller, "Fibre bundles" , McGraw-Hill (1966) {{MR|0229247}} {{ZBL|0144.44804}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> J.C. Moore, "On the periodicity theorem for complex vector bundles" , ''Sem. H. Cartan'' (1959–1960)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> M.F. Atiyah, R. Bott, "On the periodicity theorem for complex vector bundles" ''Acta Math.'' , '''112''' (1964) pp. 229–247 {{MR|0178470}} {{ZBL|0131.38201}} </TD></TR></table>
  
 
====Comments====
 
====Comments====
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Bott, "Lectures on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017220/b01722033.png" />" , Benjamin (1969) {{MR|0258020}} {{ZBL|0194.23904}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> M. Karoubi, "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017220/b01722034.png" />-theory" , Springer (1978) {{MR|0488029}} {{ZBL|0382.55002}} </TD></TR></table>
+
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Bott, "Lectures on $K(X)$" , Benjamin (1969) {{MR|0258020}} {{ZBL|0194.23904}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> M. Karoubi, "$K$-theory" , Springer (1978) {{MR|0488029}} {{ZBL|0382.55002}} </TD></TR></table>

Latest revision as of 13:07, 24 December 2020


A fundamental theorem in $ K $- theory which, in its simplest form, states that for any (compact) space $ X $ there exists an isomorphism between the rings $ K(X) \otimes K(S ^ {2} ) $ and $ K(X \times S ^ {2} ) $. More generally, if $ L $ is a complex vector bundle over $ X $ and $ P(L \oplus 1) $ is the projectivization of $ L \oplus 1 $, then the ring $ K(P(L \oplus 1)) $ is a $ K(X) $- algebra with one generator $ [H] $ and a unique relation $ ([H] - [1])([L][H] - [1]) = 0 $, where $ [E] $ is the image of a vector bundle $ E $ in $ K(X) $ and $ H ^ {-1} $ is the Hopf fibration over $ P(L \oplus 1) $. This fact is equivalent to the existence of a Thom isomorphism in $ K $- theory for complex vector bundles. In particular, $ P(1 \oplus 1) = X \times S ^ {2} $. Bott's periodicity theorem was first demonstrated by R. Bott [1] using Morse theory, and was then re-formulated in terms of $ K $- theory [6]; an analogous theorem has also been demonstrated for real fibre bundles.

Bott's periodicity theorem establishes the property of the stable homotopy type of the unitary group $ U $, consisting in the fact that $ {\Omega ^ {2} } U \sim U $, where $ \Omega X $ is the space of loops on $ X $, and $ \sim $ is weak homotopy equivalence, in particular $ \pi _ {i} (U) = \pi _ {i+2} (U) $ for $ i = 0, 1 \dots $ where $ \pi _ {i} $ is the $ i $- th homotopy group. Similarly, for the orthogonal group $ O $:

$$ \Omega ^ {8} O \sim O,\ \ \pi _ {i} (O) = \pi _ {i+ 8 } (O). $$

References

[1] R. Bott, "The stable homotopy of the classical groups" Ann. of Math. (2) , 70 : 2 (1959) pp. 313–337 MR0110104 Zbl 0129.15601
[2] J.W. Milnor, "Morse theory" , Princeton Univ. Press (1963) MR0163331 Zbl 0108.10401
[3] M.F. Atiyah, "$K$-theory: lectures" , Benjamin (1967) MR224083
[4] D. Husemoller, "Fibre bundles" , McGraw-Hill (1966) MR0229247 Zbl 0144.44804
[5] J.C. Moore, "On the periodicity theorem for complex vector bundles" , Sem. H. Cartan (1959–1960)
[6] M.F. Atiyah, R. Bott, "On the periodicity theorem for complex vector bundles" Acta Math. , 112 (1964) pp. 229–247 MR0178470 Zbl 0131.38201

Comments

References

[a1] R. Bott, "Lectures on $K(X)$" , Benjamin (1969) MR0258020 Zbl 0194.23904
[a2] M. Karoubi, "$K$-theory" , Springer (1978) MR0488029 Zbl 0382.55002
How to Cite This Entry:
Bott periodicity theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bott_periodicity_theorem&oldid=46124
This article was adapted from an original article by A.F. Shchekut'ev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article